Partitioning 3-homogeneous latin bitrades

A latin bitrade $(T^{\diamond },T^{\otimes })$ is a pair of partial latin squares which defines the difference between two arbitrary latin squares $L^{\diamond }\supseteq T^{\diamond }$ and $L^{\diamond }\supseteq T^{\otimes }$ of the same order. A 3-homogeneous bitrade $(T^{\diamond },T^{\otimes })$ has three entries in each row, three entries in each column, and each symbol appears three times in $T^{\diamond }$. Cavenagh (2006) showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh’s result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations of spherical, euclidean or hyperbolic space.